Provided herein are photonic devices configured to display photonic band gap structure with a degenerate or a split band edge. electromagnetic radiation incident upon these photonic devices can be converted into a frozen mode characterized by a significantly increased amplitude, as compared to that of the incident wave. The device can also be configured as a resonance cavity with a giant transmission band edge resonance. In an exemplary embodiment, the photonic device is a periodic layered structure with each unit cell comprising at least two anisotropic layers with misaligned anisotropy. The degenerate or split band edge at a given frequency can be achieved by proper choice of the layers' thicknesses and the misalignment angle. In another embodiment, the photonic device is configured as a waveguide periodically modulated along its axis.
|
1. A photonic device, comprising:
a periodic structure having a plurality of periodic segments, wherein individual periodic segments comprise at least two anisotropic layers with misaligned anisotropy, the periodic structure having an electromagnetic dispersion relation that exhibits a frequency gap with a split band edge for an input electromagnetic wave propagation in a first direction, wherein the structure is periodic in the first direction and wherein 100% of the incident electromagnetic radiation of the input electromagnetic wave propagates in the first direction regardless of a polarization of the incident electromagnetic radiation.
19. A photonic system, comprising:
a photonic device configured to display a split band edge, the photonic device comprising a first end, a second end, a first surface located on the first end and a plurality of segments coupled together between the first and second ends, each segment comprising:
a first anisotropic layer;
a second anisotropic layer misaligned with the first anisotropic layer; and
a third layer, wherein the photonic device is configured to convert 100% of the electromagnetic radiation of an input electromagnetic wave incident on the first surface into a frozen mode when the electromagnetic wave operates at a frequency in proximity with the split band edge regardless of a polarization of the incident electromagnetic radiation.
25. A photonic system, comprising:
a photonic device configured to display a split band edge, the photonic device comprising a first end, a second end, a first surface located on the first end and a plurality of periodic segments coupled together between the first and second ends, each segment comprising:
a first anisotropic layer having a first thickness; and
a second anisotropic layer misaligned with the first anisotropic layer and having a second thickness different from the first thickness;
wherein the photonic device is configured to convert 100% of the electromagnetic radiation of an input electromagnetic wave incident on the first surface into a frozen mode when the electromagnetic wave operates at a frequency in proximity with the split band edge regardless of a polarization of the electromagnetic radiation.
2. The photonic device of
3. The photonic device of
4. The photonic device of
5. The photonic device of
6. The photonic device of
7. The photonic device of
9. The photonic device of
10. The photonic device of
11. The photonic device of
12. The photonic device of
13. The photonic device of
14. The photonic device of
15. The photonic device of
16. The photonic device of
17. The photonic device of
20. The photonic system of
21. The photonic system of
22. The photonic device of
23. The photonic system of
24. The photonic system of
26. The photonic system of
27. The photonic system of
28. The photonic device of
29. The photonic system of
30. The photonic system of
|
This application is a continuation of U.S. application Ser. No. 12/131,897 filed Jun. 2, 2008, now abandoned which claims the benefit of U.S. Provisional application Ser. No. 60/941,055 filed May 31, 2007, and is a continuation-in-part of U.S. application Ser. No. 13/019,121 filed Feb. 1, 2011, which is a continuation of U.S. application Ser. No. 11/720,592 filed May 31, 2007, now U.S. Pat. No. 7,881,570, which claims the benefit of U.S. Provisional Application Ser. No. 60/648,319 filed Jan. 28, 2005, the specification of which applications are incorporated herein by reference.
This invention was made with Government support under grant number F49620-01-1-0567, awarded by Air Force Office of Scientific Research, Air Force Materials Command, USAF. The Government has certain rights to this invention.
The invention relates generally to the field of photonic devices, and more particularly to systems and methods for transmitting and storing electromagnetic radiation in a photonic device with inhomogeneous spatially periodic structure.
The manipulation of electromagnetic energy can be advantageous to numerous applications within many industries. For instance, much effort has been focused on reducing the velocity of electromagnetic energy, such as light and microwave pulses. The reduced velocity of electromagnetic energy can facilitate manipulation of electromagnetic waves. It can also enhance the light-matter interaction essential in numerous optical and microwave applications. One approach to reducing the electromagnetic energy velocity is through the use of spatially inhomogeneous periodic media displaying strong spatial dispersion at operational frequencies. Spatial inhomogeneity results in strong nonlinear relation between the frequency ω of propagating electromagnetic wave and the respective Bloch wave number k. The relation ω(k) is referred to as dispersion relation or, equivalently, as k−ω diagram. At certain frequencies, the wave group velocity v=dω/dk vanishes implying extremely low energy velocity.
One common photonic device exploiting spatial inhomogeneity is a photonic crystal. This device is typically composed of multiple repeating segments (unit cells) arranged in a periodic manner. Electromagnetic frequency spectrum of a typical photonic crystal develops frequency bands separated by forbidden frequency gaps. The frequency separating a photonic band from adjacent photonic gap is referred to as a (photonic) band edge, or simply a band edge. At frequencies close to a photonic band edge, the relationship between the frequency ω and the wave number k can be approximated as
′ω−ωg∝(k−kg)2, (1)
implying that the respective group velocity
v=dω/dk∝k−kg∝√{square root over (ω−ωg)} (2)
vanishes as ω approaches the band edge frequency ωg. This creates conditions for very slow pulse propagation. Another common photonic device exploiting spatial inhomogeneity and providing conditions for slow energy propagation is a periodic array of weakly coupled resonators. There exist many different physical realizations of the individual resonators connected into the periodic chain.
One common drawback of current photonic devices employing spatial inhomogeneity is that only a small fraction of the incident electromagnetic radiation is converted into the slow electromagnetic mode, resulting in low efficiency of the device. Another common drawback of current photonic devices is the necessity to employ a large number of the said segments (unit cells) in order to achieve a desirable slowdown of electromagnetic energy. Accordingly, improved photonic devices are needed having smaller dimensions and allowing for more efficient manipulation of the incident electromagnetic radiation.
The devices, systems and methods described in this section are done so by way of exemplary embodiments that are not intended to limit these devices, systems and methods in any way.
In one exemplary embodiment, a photonic system is provided that includes a photonic device configured to display a degenerate band edge, the photonic device including a first end, a second end, a first surface located on the first end and a plurality of segments coupled together between the first and second ends. Each segment can include a first anisotropic layer, a second anisotropic layer misaligned with the first anisotropic layer, and a third layer. The photonic device can be configured to convert an electromagnetic wave incident on the first surface into a frozen mode, where the electromagnetic wave operates at a frequency in proximity with the degenerate band edge.
In another exemplary embodiment, a photonic system is provided that includes a photonic device configured to display a degenerate band edge, the photonic device including a first end, a second end, a first surface located on the first end and a plurality of periodic segments coupled together between the first and second ends. Each segment can include a first anisotropic layer having a first thickness and a second anisotropic layer misaligned with the first anisotropic layer and having a second thickness different from the first thickness. The photonic device can be configured to convert an electromagnetic wave incident on the first surface into a frozen mode, when the electromagnetic wave operates at a frequency in proximity with the degenerate band edge.
In another exemplary embodiment, a photonic system is provided that includes a photonic device configured to display a split band edge, the photonic device including a first end, a second end, a first surface located on the first end and a plurality of segments coupled together between the first and second ends. Each segment can include a first anisotropic layer, a second anisotropic layer misaligned with the first anisotropic layer, and a third layer. The photonic device can be configured to convert an electromagnetic wave incident on the first surface into a frozen mode, where the electromagnetic wave operates at a frequency in proximity with the split band edge.
In another exemplary embodiment, a photonic system is provided that includes a photonic device configured to display a split band edge, the photonic device including a first end, a second end, a first surface located on the first end and a plurality of periodic segments coupled together between the first and second ends. Each segment can include a first anisotropic layer having a first thickness and a second anisotropic layer misaligned with the first anisotropic layer and having a second thickness different from the first thickness. The photonic device can be configured to convert an electromagnetic wave incident on the first surface into a frozen mode, when the electromagnetic wave operates at a frequency in proximity with the split band edge.
Other systems, methods, features and advantages of the invention will be or will become apparent to one with skill in the art upon examination of the following figures and detailed description. It is intended that all such additional systems, methods, features and advantages be included within this description, be within the scope of the invention, and be protected by the accompanying claims. It is also intended that the invention not be limited to the details of the example embodiments.
The details of the invention, including fabrication, structure and operation, may be gleaned in part by study of the accompanying figures, in which like reference numerals refer to like parts. The components in the figures are not necessarily to scale, emphasis instead being placed upon illustrating the principles of the invention. Moreover, all illustrations are intended to convey concepts, where relative sizes, shapes and other detailed attributes may be illustrated schematically rather than literally or precisely.
Photonic devices and systems having degenerate or split spectral band edges and methods for using the same are described herein. These devices, systems and methods are based on the physical idea of using spatially periodic structures displaying a degenerate band edge
ω−ωd∝(k−kd)4, (3)
rather then the regular band edge described by equation (1). Unlike the regular band edge (1), display of the degenerate band edge (3) allows for the frozen mode regime, accompanied by a complete conversion of the incident radiation into a slow mode with a drastically enhanced amplitude. In addition, a resonance cavity incorporating a photonic device displaying a degenerate band edge can have much smaller relative dimensions compared to those incorporating existing photonic devices.
Light transmitting periodic structures that can be configured to display the degenerate band edge (3) include, but are not limited to: (i) photonic crystals, such as periodic layered structures, as well as structures with two and three dimensional periodicity, (ii) spatially modulated optical and microwave waveguides and fibers, and (iii) arrays of coupled resonators. The embodiments discussed below are directed towards periodic arrays of anisotropic dielectric layers; however, it is important to emphasize that the underlying reason for the enhanced performance of the photonic device as described herein lies in the existence of a degenerate band edge (3) in the respective frequency spectrum. Specific physical realization of the periodic structure displaying such a spectrum is determined by practical needs, i.e., one of ordinary skill in the art will readily recognize how to implement spatially modulated optical and microwave waveguides and fibers, arrays Of coupled resonators and other desired structural configurations based on the embodiments described herein.
In this embodiment, the structure of photonic device 101 is periodic along the Z direction perpendicular to layers 106-108, which are parallel to the X-Y plane. The X, Y and Z directions are perpendicular to each other like that of a standard Cartesian coordinate system. Photonic device 101 is also preferably homogeneous in the in-plane directions X and Y, although photonic device 101 can also be inhomogeneous in the directions X, Y, or both, if desired. The total number N of repeating segments 105 in photonic device 101 depends on the specific application and usually varies between three and several hundred, although device 101 is not limited to this range of segments 105.
The anisotropy axes of anisotropic layers 106 and 107 preferably have misaligned orientation in the X-Y plane with the misalignment angle φ being different from 0 and π/2. In this embodiment, anisotropic layers 106 and 107 are composed of the same anisotropic dielectric material and have a variable misalignment angle. The dielectric permittivity tensors of the three constitutive layers 106, 107 and 108 can be chosen as follows:
where ∈A1, ∈A2 and ∈B, are the dielectric permittivity tensors for the layers 106, 107 and 108, respectively. The choice (6) for the material tensor ∈B corresponds to the case where layer 108 is an empty gap between the adjacent pairs of anisotropic layers 106 and 107. If desired, optional layer 108 can also be filled with either anisotropic or isotropic material, such as glass, air, active or nonlinear medium, etc., or it can be left vacant (e.g., as a vacuum), depending on the specific practical needs of the application. The quantity δ in (6) describes inplane anisotropy of the A-layers 106 and 107, essential for the existence of degenerate band edge. The parameter φ in (6) designates the misalignment angle between anisotropic layers 106 and 107. It can be chosen anywhere between 0 and π, which provides additional tunability of the photonic device. The k−ω diagram of the photonic device in
Otherwise, the characteristics of this embodiment in
In one exemplary embodiment, each segment 605 can be subdivided into three adjacent portions 606-608, as depicted in
Again, waveguide 101 as depicted in
The input electromagnetic wave 602 enters waveguide 101 in
If this embodiment of waveguide 101 is empty, or if it is filled with a uniform dielectric substance, the misaligned cross-section anisotropy of portions 606 and 607 may become required for the existence of degenerate band edge. But if the filling material is not distributed uniformly, as shown in the exemplary embodiments of
In the exemplary embodiment depicted in
There can be a practically infinite number of specific waveguide realizations of photonic device 101 displaying the degenerate band edge (3). In the embodiment in
Described below are three exemplary methods in which photonic device 101 can be used. Each of the methods is referred to as an independent regime. It should be noted that operation in any one regime is dependent on the needs of the specific application, and that these regimes do not constitute an exhaustive list of potential uses for photonic device 101. In fact, photonic device 101 can be operated in any one or more of these three regimes as well as other regimes not explicitly described herein.
A first exemplary regime for photonic device 101 can be referred to as the frozen mode regime at degenerate band edge frequency. In this regime, an incident electromagnetic pulse 401 with central frequency close to that of the degenerate band edge is transmitted to photonic device 101, where it is converted into the frozen mode 402 having greatly enhanced amplitude and compressed length, similar to the exemplary embodiment described with respect to
A second exemplary regime for photonic device 101 can be referred to as the abnormal surface wave near degenerate band edge frequency regime. In the embodiments in
A∝NA0, (4)
where A0 is the amplitude of the incident plane wave. Strong cavity resonance can require a large number N of unit cells (e.g., segments 105, 605 and 705) in the periodic structure. A similar resonance effect occurs in periodic structures in the vicinity of the degenerate photonic band edge (3), as depicted in
A∝N2A0, (5)
and referred to as the giant transmission band edge resonance. This shows that a Fabry-Perot cavity based on photonic device 101 configured to display the degenerate band edge is much more efficient than previous versions. For instance, a resonance cavity with the degenerate band edge based on an exemplary embodiment of device 101 having 10 periodic segments (e.g, 105, 605 or 705), or unit cells, can perform as well as a regular Fabry-Perot photonic cavity composed of 100 periodic segments. The optimal number of segments (e.g, 105, 605 or 705) in the resonance cavity depends on specific application, but in any event, cavities based on photonic device 101 configured to display the degenerate band edge can be much smaller.
The embodiments described with respect to
Photonic device 101 can be incorporated in numerous photonic systems implemented in a myriad of applications. For instance, photonic device 101 can be implemented as a tunable delay line, an efficient nonlinear element used for frequency conversion, wave mixing and the like, it can also be used as a high performance resonance cavity in an optical amplifier and in a laser, a host for a multi-dimensional optical network, an incident wave receiver and the like. It should be noted that these examples are not intended to limit, in any way, the systems and methods in which photonic device 101 can be used. Nor are these examples intended to limit photonic device 101 to any one type of system, application or technology.
Photonic Devices and Systems having Split Spectral Band Edge
A more detailed discussion of photonic devices and systems having split spectral band edges and methods for using the same are described below.
Every single implementation and device mentioned above can be slightly modified so that instead of a DBE, the respective dispersion relation displays a split photonic band edge (SBE). The modification may involve, but is not limited to, a small change in structural geometry of the periodic array and/or small change in physical characteristics of the materials involved, and/or small change of the angle of wave incidence on the boundary of the periodic structure. The exact nature of the modification is not important, because almost any small modification can transform a DBE to SBE. The functionality and performance of the modified device would remain basically the same except for one critical improvement. Namely, a SBE based device can utilize virtually 100% of the incident electromagnetic radiation regardless of its polarization. By comparison, the DBE based devices described above can utilize only one single polarization component of the incident electromagnetic wave, while the other polarization component is reflected back to space.
Another modification which can result in significant performance enhancement is positioning of a reflector (a mirror) next to the output boundary of the photonic device.
Physical Background
Wave propagation in spatially periodic media, such as photonic crystals, can be qualitatively different from any uniform substance. The differences are particularly pronounced when the wavelength is comparable to the primitive translation L of the periodic structure. The effects of strong spatial dispersion culminate at stationary points ωs=ω(ks) of the Bloch dispersion relation where the group velocity u=∂ω/∂k of a traveling Bloch wave vanishes
One reason for this is that vanishing group velocity always implies a dramatic increase in density of modes at the respective frequency. In addition, vanishing group velocity also implies certain qualitative changes in the eigenmode structure, which can be accompanied by some spectacular effects in wave propagation. A particular example of the kind is the frozen mode regime associated with a dramatic amplitude enhancement of the wave transmitted to the periodic medium [2-7, 9]. In this specification, the focus is on a different slow-wave effect, namely, on a Fabry-Perot resonance in a plane-parallel photonic-crystal cavity. This phenomenon is also referred to as the transmission band edge resonance. There are some similarities between the frozen mode regime and the slow-wave cavity resonance in a plane-parallel photonic crystals. Both effects are associated with vanishing group velocity at a stationary point (1) of the dispersion relation. As a consequence, both effects are strongly dependent on specific type of spectral singularity (1). A fundamental difference though is that the frozen mode regime is not a resonance phenomenon in a sense that it is not particularly sensitive to the shape and size of the photonic crystal. For instance, the frozen mode regime can occur even in a semi-infinite periodic structure, where the incident plane wave is converted to a frozen mode slowly propagating through the periodic medium until it is absorbed. By contrast, in the case of a photonic-crystal cavity resonance, the entire bounded periodic structure acts as a resonator, resulting in a strong sensitivity of the resonance behavior to the size and shape of the photonic crystal.
Photonic-crystal cavity resonance has numerous and diverse practical applications. This slow-wave phenomenon is widely used for the enhancement of light-matter interactions such as nonlinear and nonreciprocal effects, optical activity, light amplification and lasing, etc. It can also be used in optical and microwave filters, delay lines, as well as for the enhancement of antenna gain and directionality. Our objective is to design a photonic crystal cavity with drastically reduced dimensions and enhanced performance, compared to that of a common Fabry-Perot resonator based on a periodic stack of non-birefringent layers. The idea is to employ periodic structures supporting the dispersion relations different from those allowed in periodic arrays of non-birefringent layers. Indeed, periodic arrays involving birefringent layers can display stationary points (1) different from a regular photonic band edge. Some examples are shown in
WDBE
∝WIN4, (2)
where WI is the energy density of the incident wave and N is the total number of unit cells in the periodic stack. By comparison, the energy density at a regular transmission band-edge resonance in
WRBE
∝WIN2, (3)
The estimations (2) and (3) imply that the Q-factor of a cavity with a DBE can be by factor N2 higher compared to that of a regular cavity of the same size N. This is a huge difference.
Periodic structures involving birefringent layers have one fundamental problem. Namely, their reflectance and transmittance are essentially dependent on the incident wave polarization. This dependence is particularly strong near the edge of transmission band, where the cavity resonances occur. At the respective frequency range, a periodic stack with birefringent layers acts as a polarizer, reflecting back to space at least half of the incident wave energy. As a consequence, a DBE cavity works only with one (elliptic) polarization component of the incident wave, while the other polarization component is reflected by the periodic structure. Similar fundamental problem exists in all different modifications of the frozen mode regime considered in [3, 4, 6, 7, 9]. For many applications, such a polarization sensitivity may not be acceptable, and, thus, a way to utilize 100% of the incident wave energy needs to be found. In this specification, one of skill in the art is shown how to solve the polarization problem by slightly modifying the periodic layered array, so that instead of a DBE the respective spectral branch displays a split photonic band edge (SBE) shown in
Similar approach can be applied not only to a PCCR, but also to all different modifications of the frozen mode regime described in [3, 4, 6, 7, 9].
Photonic-Crystal Cavity Resonance (PCCR) in a Periodic Stack of Anisotropic Layers
Dispersion Relation in Periodic Arrays of Birefringent Layers
For simplicity, the case is restricted to a plane monochromatic wave normally incident on a plane-parallel periodic layered structure. Inside the periodic medium, the electromagnetic field is a superposition of Bloch eigenmodes Ψk(z) defined by the following equality
Ψk(z÷L)=Ψk(z)exp(ikL). (4)
where L is the primitive translation. The direction z of wave propagation is normal to the layers. The Bloch wave number k in (4) is defined up to a multiple of 2πr/L. Real k correspond to propagating (traveling) Bloch modes, while complex k correspond to evanescent modes. Evanescent modes can be relevant only near photonic crystal boundary, or some other defects of the periodic structure. The correspondence between the frequency ω and the Bloch wave number k is referred to as the Bloch dispersion relation. In reciprocal and/or centrosymmetric periodic structures, the Bloch dispersion relation is always symmetric with respect to the points k=0 and k=π/L of the Brillouin zone
ω(k0+k)=ω(k0−k), (5)
where
k0=0, π/L. (6)
Typical examples of k−ω diagrams of periodic layered arrays are shown in
ω(k)≠ω(−k). (7)
Examples of layered structures with asymmetric dispersion relation (7) can be found in [2, 3]. This specification only discusses non-magnetic structures, unless otherwise explicitly stated.
Every spectral branch of the k−ω diagram displays stationary points (1) associated with vanishing group velocity. Usually, such points are located at the center and at the boundary of the Brillouin zone
ks=k0=0, π/L. (8)
This is always the case in periodic layered structures composed of non-birefringent layers, where all stationary points coincide with photonic band edges. If, on the other hand, some of the layers in a unit cell are birefringent, then in addition to (8), the dispersion relation can also develop a reciprocal pair of stationary points corresponding to a general value of the Bloch wave number k, as shown in
Geometrical Condition for Photonic-Crystal Cavity Resonance
Consider a finite plane-parallel periodic structure composed of N unit cells L. If the number N is significant, the electromagnetic properties of the periodic stack can be effectively described in terms of the Bloch eigenmodes (4) of the corresponding infinite periodic layered medium. For instance, a periodic stack having as few as several periods L can display almost total reflectivity at photonic band gap frequencies, which reflects the absence of propagating Bloch modes at the respective frequency range.
A typical dependence of finite stack transmittance on the incident wave frequency is shown in [8] and [1]. The series of sharp peaks in the transmission band corresponds to photonic-crystal cavity resonances (PCCR). The resonance frequencies are located close to a photonic band edge, where the Bloch waves have very low group velocity. At each resonance m, the electromagnetic field Ψ(z) inside the periodic stack is close to a standing wave composed of a pair of reciprocal Bloch waves
ΨT(z)=Ψk
with large and nearly equal amplitudes and low group velocity. The photonic crystal boundaries coincide with the standing wave nodes, where the forward and backward Bloch waves interfere destructively to meet the boundary conditions. The latter circumstance determines the wave numbers of the forward and backward Bloch components at the resonance frequencies
where k0 is defined in (6). The integer m denotes the resonance peaks in order of their distance from the photonic band edge. The respective resonance frequencies are expressed in terms of the dispersion relation
ωm=ω(km). (11)
The proximity of PCCR to the photonic band edge is determined by the number N of unit cells in the periodic stack. The expressions (10) and (11) only apply if
N>>m. (12)
The following dimensionless notations are introduced for the small deviation of the wave number and the frequency from the respective stationary point
κ=|k−k0|L, v(κ)=[ω(k)−ω(k0)]L/c. (13)
According to (10) and (11), the resonance values of k and v are
The most powerful resonance is usually the one closest to the respective photonic band edge
PCCR in the Vicinity of SBE
The relations (9) through (15) describe the resonance conditions for a single pair (9) of reciprocal Bloch waves. In periodic stacks involving birefringent layers, a PCCR produced by a single pair of reciprocal. Bloch waves can be excited only by one polarization component of the incident plane wave. This implies that if the incident wave polarization is random, then at least half of the incident light energy will not enter the periodic medium and will be reflected back to space by the photonic crystal boundary. For instance, this scenario always occurs at DBE related giant PCCR. At certain frequencies, though, there can be two pairs of reciprocal Bloch waves with different wave numbers; each pair having its own polarization. Of particular interest here is the frequency range
ω0<ω<ωb (16)
of the k−ω diagram in
To start with, the dispersion curve is considered with a SBE in
where
a/b<0 (18)
and
|a/b|<<1. (19)
The inequality (18) is the condition for SBE. Indeed, in the opposite case of
a/b>0, (20)
the dispersion curve (17) would develop a RBE at k=0, as shown in
a/b=0, (21)
the dispersion curve (17) would have a DBE at k=0, as shown in
There are three stationary points associated with SBE in
κa=0, va=0. (22)
It is located either at the center of the Brillouin zone, or at its boundary. The other two stationary points correspond to the actual SBE
±κb=±√{square root over (−a/b)}, vb=−a2/4b. (23)
Taking into account (23), the condition (19) for the proximity of the SBE to a DBE can be recast as
κb<<1. (24)
The condition (24) implies that the points b1 and b2 on the dispersion curve are close to each other.
In what follows, for simplicity it is assumed that
b<0<a. (25)
In this case, the SBE in question corresponds to the upper edge of the transmission band, as shown in
a<0<b (26)
corresponds to the SBE being the lower edge of the transmission band. There is no qualitative difference between the two cases.
At any given frequency v within the range
va<v<vb, (27)
there are two pairs of reciprocal Bloch wave. Each pair comprises one forward and one backward propagating modes with equal and opposite wave numbers and group velocities
The pair of wave numbers (28) corresponds to the concave portion of the dispersion curve (17), while the pair of wave numbers (29) corresponds to the convex portion of the dispersion curve. Obviously,
|κin|<|κex|, at va<v<vb. (30)
Double PCCR
Within the frequency range (27), either pair of the reciprocal Bloch waves (28) and (29) can develop a PCCR. Of particular interest here is the situation where a PCCR associated with one pair of the reciprocal Bloch waves occurs at the same or almost the same frequency as that of the other pair of the reciprocal Bloch waves. This situation is referred to as a double PCCR.
Starting with case (28) of the reciprocal pair of Bloch waves corresponding to the concave section of the dispersion curve, it is possible that the frequency range (27) contains only a single PCCR—the one with m=1. According to (23) and (10), the condition for a single PCCR is
The respective resonance frequency vr is determined by (11) and (17)
where k1=π/N.
Consider now a PCCR associated with the reciprocal pair (29) of Bloch waves corresponding to the convex section of the dispersion curve. The condition that the latter PCCR occurs at the same frequency (32) as the PCCR related to the concave section (28) of the dispersion curve is imposed
Simple analysis shows that it is only possible if at v=vr the following is given
The relation
kex=2kin
together with (28) and (29) yield
The frequency of the double PCCR is
The group velocities of the two reciprocal pairs of Bloch waves at the resonance frequency vr are
By comparison, in the case of DBE related PCCR, the following estimations are provided for the resonance frequency v1 and the respective group velocity of the two Bloch components
which are similar to the case of double PCCR at SBE. This similarity proves that both DBE-related giant PCCR and SBE-related double PCCR produce the same field enhancement (2). At the same time, our numerical analysis shows that the SBE-related double PCCR provides 100% conversion of the incident wave into the resonance field inside the periodic stack. By contrast, in the case of DBE-related giant PCCR, more than half of the incident wave energy is reflected back to space even before in enters the periodic medium.
Slow-Wave Resonance in Periodic Stacks of Anisotropic Layers
A detailed discussion of photonic devices and systems and methods for using the same according to various embodiments of the invention are described below.
A Fabry-Perot resonance (a transmission band edge resonance) in periodic layered structures involving birefringent layers is considered. In a previous publication [Phys. Rev. E 72, 036619 (2005)] it has been shown that the presence of birefringent layers with misaligned in-plane anisotropy can dramatically enhance the performance of the photonic-crystal resonator. It allows us to reduce its size by an order of magnitude without compromising on its performance. The key characteristic of the enhanced slow-wave resonator is that the Bloch dispersion relation ω(k) of the periodic structure displays a degenerate photonic band edge, in the vicinity of which the dispersion curve can be approximated as Δω˜(Δk)4, rather than Δω˜(Δk)2. Such a situation can be realized in specially arranged stacks of misaligned anisotropic layers. On the down side, the presence of birefringent layers results in the slow-wave resonance being coupled only with one (elliptic) polarization component of the incident wave, while the other polarization component is reflected back to space. Here, it is shown how a small modification of the periodic layered array can solve the above fundamental problem and provide a perfect impedance match regardless of the incident wave polarization, while preserving the giant slow-wave resonance characteristic of a degenerate photonic band edge. Both features are of critical importance for many practical applications, such as the enhancement of various light-matter interactions, light amplification and lasing, optical and microwave filters, antennas, etc.
Physical Background
Wave propagation in spatially periodic media, such as photonic crystals, can be qualitatively different from any uniform substance. The differences are particularly pronounced when the wavelength is comparable to the primitive translation L of the periodic structure [11-17]. The effects of strong spatial dispersion culminate at stationary points ωs=ω(ks) of the Bloch dispersion relation where the group velocity u=≢ω/∂k of a traveling Bloch wave vanishes
One reason for this is that vanishing group velocity always implies a dramatic increase in density of modes at the respective frequency. In addition, vanishing group velocity also implies certain qualitative changes in the eigenmode structure, which can be accompanied by some spectacular effects in wave propagation. A particular example of the kind is the frozen mode regime associated with a dramatic amplitude enhancement of the wave transmitted to the periodic medium [18-23]. In this specification, the focus is on a different slow-wave effect, namely, on a Fabry-Perot resonance in bounded photonic crystals. This slow wave phenomenon, illustrated in
It is also important to distinguish between two qualitatively different classes of photonic-crystal resonators. The first class comprises resonance cavities where the role of periodic dielectric structure reduces to electromagnetic (EM) field confinement by reflecting it back to the cavity interior. The resonance frequency (or frequencies) of such photonic cavities usually lies in a frequency gap (a stop band) of the photonic crystal. The periodic dielectric array here plays the role of a distributed Bragg reflector. The number of resonance modes depends on the cavity size. It can be a single mode localized on an isolated defect inside the photonic crystal [24,25]. Or the cavity can support multiple resonances, if its size significantly exceeds the light wavelength. More detailed information on photonic crystal cavities can be found in numerous papers and monographs on optics and photonics (see, for example, Ref. [26], and references therein). In this specification, the subject will not be discussed further.
The second class of photonic-crystal resonators comprises the so-called slow-wave resonators. They are qualitatively different from the band-gap cavities. In slow-wave photoniccrystal resonators, the reflectors may not be needed at all, as shown in the example in
In the specification, a slow-wave photonic-crystal resonator is described with drastically reduced dimensions and enhanced performance, compared to that of a common Fabry-Perot resonator based on a periodic stack of nonbirefringent layers. The idea is to employ periodic structures supporting the dispersion relations different from those allowed in periodic arrays of nonbirefringent layers. Indeed, periodic arrays involving birefringent layers can display stationary points (1) different from a regular photonic band edge in
WDBE
∝WIN4. (2)
where WI is the energy density of the incident wave and N is the total number of unit cells in the periodic stack. By comparison, the average EM energy density at a regular trans-mission band-edge resonance in
WRBE
∝WIN2. (3)
The estimations (2) and (3) imply that the Q factor of a DBE-based slow-wave resonator can be by factor N2 higher compared to that of a RBE related Fabry-Perot resonator of the same size; this a huge difference. A detailed comparative analysis of the giant DBE related slow-wave resonance versus the regular transmission band edge resonance can be found in Ref. [33].
On the down side, periodic structures with birefringent layers have a fundamental problem—their reflectance and transmittance are essentially dependent on the incident wave polarization. This dependence is particularly strong near the edges of transmission bands, where the slow-wave resonances occur. In particular, a DBE related giant transmission resonance described in Ref. [33] is coupled only with one (elliptic) polarization component of the incident wave, while the other polarization component is reflected back to space by the photonic crystal boundary [33]. In other words, at the resonance frequency, a periodic stack involving birefringent layers acts as a polarizer, reflecting back to space roughly half of the incident wave energy. This behavior is illustrated in
Geometrical Description of Slow-Wave Resonance
Steady-state Fabry-Perot resonance in a plane-parallel photonic crystal is commonly described as a standing Bloch wave composed of a pair of reciprocal propagating Bloch modes with equal and opposite wave numbers and group velocities
ΨT(z)=Ψk(z)+Ψ−k(z). 0<z<D. (4)
At a resonance, the two propagating Bloch components in Eq. (4) have large and nearly equal amplitudes and low group velocities. Two nodes of the standing wave coincide with the photonic crystal boundaries at z=0 and z=D, which determines the resonance values km of the Bloch wave number k
where k0 coincides with one of stationary points (1) of the dispersion relation ω(k) (usually, k0=0 or Ir). The integer m denotes the resonance peaks in order of their distance from the respective photonic band edge at k0. The resonance frequencies ωm are expressed in terms of the Bloch dispersion relation
ωm=ω(km). (6)
and located close to the photonic band edge at ω0=ω(k0). According to Eq. (5), the proximity of the resonances to the photonic band edge is determined by the number N of unit cells in the periodic stack. The expressions (5) and (6) only apply if
N>>m. (7)
The simple representation (4) for the resonance field distribution works very well in the vicinity of a regular photonic band edge (RBE). In the case of a giant transmission resonance associated with a degenerate photonic band edge (DBE), the above simple picture does not apply. Indeed, according to Ref. [33], in the vicinity of a DBE, the evanescent mode contribution becomes equally important, and the resonance field ΨT(z) cannot be viewed as a simple standing Bloch wave (4). Yet another qualitatively different situation can occur in the vicinity of a split photonic band edge (SBE), provided that the magnitude of the split between the edges b1 and b2 in
ΨT(z)=ΨA(z)+ΨB(z). 0<z<D. (8)
where
ΨA(z)=Ψk
and
ΨB(z)=Ψk
Either a DBE or a SBE can only exist in periodic stacks involving misaligned birefringent layers, example of which is shown in
For simplicity, the case is restricted a plane monochromatic wave normally incident on a layered structure, as shown in
Transmission Band-Edge Resonance in the Vicinity of a Split Photonic Band Edge
Consider the vicinity of a split photonic band edge (SBE) on the k−ω diagram in
ω0<ω<ωb. (11)
covers a narrow portion of the transmission band which includes the SBE. At any given frequency from Eq. (11), there are two pairs of reciprocal Bloch waves with very low group velocity and different polarizations; each pair being capable of producing its own slow-wave cavity resonance with the resonance conditions similar to Eq. (5). Our focus is on the possibility of the two resonances occurring at the same frequency. Such a situation will be referred to as the double resonance. It turns out that the double transmission band edge resonance is as powerful as the giant resonance associated with a DBE. However, in addition, a SBE related resonance utilizes all the energy of the incident wave regardless of its polarization. By contrast, a DBE based giant transmission resonance is coupled only with one polarization component of the incident wave; the rest of the incident wave energy being reflected back to space. This important difference is obvious if the DBE related transmission dispersion shown in
The following dimensionless notations for the small deviation of the wave number and the frequency from the respective stationary point is introduced:
κ=|k−k0|L, v(κ)=[ω(k)−ω(k0)]L/c. (12)
According to Esq. (5) and (6), the resonance values of κ and v are
The most powerful resonance is usually the one closest to the respective photonic band edge
A dispersion curve is considered with a SBE in more detail. If the split between the twin band edges b1 and b2 in
where
a/b<0 (16)
and
|a/b|<<1. (17)
The inequality (16) is the condition for SBE. Indeed, in the opposite case of
a/b>0, (18)
the dispersion curve (15) would develop a RBE at κ=0, as shown in
a/b=0, (19)
the dispersion curve (15) would have a DBE at κ=0. The additional inequality (17) is the condition for the proximity of the SBE to a DBE. This proximity allows us to use the expansion (15) in the frequency range spanning both twin edges of the SBE. More importantly, the condition (17) is essential for the phenomenon of the giant transmission resonance in the vicinity of SBE.
There are three stationary points associated with a SBE. The first one is trivial
κa=0. va=0. (20)
It is located either at the center of the Brillouin zone, or at its boundary. The other two stationary points correspond to the actual SBE
±κb=±√{square root over (−a/b)}. vb=−a2/4b. (21)
Taking into account Eq. (21), the condition (17) for the proximity of the SBE to a DBE can be recast as
κb<<1. (22)
The condition (22) implies that the points b1 and b2 on the dispersion curve are close to each other.
In what follows, it is assumed for simplicity that
b<0<a. (23)
In this case, the SBE in question corresponds to the upper edge of the transmission band, as shown in
a<0<b (24)
corresponds to the SBE being the lower edge of the respective transmission band. There is no qualitative difference between the two cases.
At any given frequency v within the range
va<v<vb. (25)
there are two pairs (9) and (10) of reciprocal Bloch waves. Each pair comprises one forward and one backward propagating modes with equal and opposite wave numbers and group velocities
The pair of wave numbers (26) corresponds to the concave portion of the dispersion curve (15), while the pair of wave numbers (27) corresponds to the convex portion of the dispersion curve. Obviously,
kA<kB at va<v<vb. (28)
Conditions for the Double SBE Resonance
Within the frequency range (25), either pair of the reciprocal Bloch waves (9) and (10) can develop a transmission resonance. Of particular interest here is the case where the two resonances occurs at the same or almost the same frequency. This situation is illustrated in
Starting with the resonance created by the reciprocal pair (9) of Bloch waves corresponding to the concave section of the dispersion curve in
The respective resonance frequency PA is determined by Esq. (6) and (15)
where κ1=π/N.
Consider now the resonance created by the reciprocal pair (10) of Bloch waves corresponding to the convex section of the dispersion curve. The following condition is imposed
vA=vB≡vr (31)
that both resonances occur at the same frequency (30). This condition leads to the following equality:
Simple analysis shows that it is only possible if at v=vr the following is given
The relation
kA=2kB
together with Esq. (26) and (27) yield
The frequency of the double transmission resonance is
The group velocities of the two reciprocal pairs of Bloch waves at the resonance frequency vr are
By comparison, in the case of a DBE related giant trans-mission band edge resonance, the following estimation is provided for the resonance frequency v1 and the respective group velocity of the two propagating Bloch components
These estimations are similar to those related to the double transmission resonance associated with a SBE. In either case, the average resonance energy density is estimated by Eq. (3), which justifies the term “giant” transmission resonance.
The entire consideration of this subsection was based on the assumption that each pair (9) and (10) of the reciprocal Bloch modes is responsible for its own individual transmission resonance, described as a standing wave (9) or (10), respectively. While the double resonance at vr is described as the situation where the frequencies vA and vB of those individual resonances merely coincide. In fact, these two transmission resonances can be treated as independent only if the respective resonance frequencies vA and vB are separated. As soon as vA and vB are close to each other, the contributions of all four propagating Bloch modes to the resonance field ΨT(z) in Eq. (8) become equally important. The latter situation persists even if the incident wave polarization correspond to the, so-called, single mode excitation regime, defined in Refs. [23,33] for the case of a semi-infinite periodic structure. In other words, the single mode excitation regime produces almost pure ΨA(z) resonance (9) or ΨB(z) resonance (10) only if their frequencies are well separated. Otherwise, the EM field ΨB(z) is a superposition of all four Bloch eigenmodes. In such a case, the resonance field ΨT(z) cannot be viewed as a simple standing wave (4) regardless of the incident wave polarization. The physical reason for such a strong hybridization is that due to the condition (22), the RBE in question is very close to a DBE. On the other hand, in the vicinity of a DBE, all four vector columns Ψk(z) in Eq. (A4) become nearly parallel to each other [23,33]. The latter circumstance excludes the possibility of exciting only one of the two pairs of the reciprocal Bloch modes (9) or (10) in the situation (31), where the resonance conditions (5) are in place for both of them simultaneously. Still, the above consideration provides a very useful guidance on the conditions for SBE related giant transmission resonance and allows to find the proper physical characteristics of the periodic structure.
The bottom line is that the SBE related giant transmission resonance is as powerful as that related to a DBE. However, in addition, the SBE related resonance provides a perfect coupling with the incident wave regardless of its polarization.
In summary, it is stressed that the remarkable features of the DBE and SBE related giant transmission resonances can be derived from such fundamental characteristics of the periodic composite medium as its electromagnetic dispersion relation. Specific details of the periodic array, such as physical characteristics of the constitutive components, or structural geometry, are only important as long as the symmetry of the periodic array is compatible with the existence of the required spectral singularities.
Electrodynamics of Stacks of Birefringent Layers
This section briefly introduces some basic notations and definitions of electrodynamics of stratified media involving birefringent layers. A detailed and consistent description of the subject, along with numerous references, can be found in Refs. [18,19,23,33], where similar notations and terminology are used. For simplicity, the discussion herein is restricted to the case of a plane monochromatic wave normally incident on a layered structure, as shown in
Time-harmonic electromagnetic field inside and outside the layered medium can be described by the vector column
where {right arrow over (E)}(z) and {right arrow over (H)}(z) are time-harmonic electric and magnetic fields. The z direction is normal to the layers. The values of Ψ at any two locations z and z′ are related by the transfer matrix T(z,z′) defined by
Ψ(z)=T(z,z′)Ψ(z′). (A2)
The elements of the transfer matrix are expressed in terms of material tensors and other physical characteristics of the stratified medium.
Let ΨI, ΨR, and ΨP be the incident, reflected, and transmitted waves, respectively, as shown in
ΨI(0)+ΨR(0)=ΨT(0), ΨP(D)=ΨT(D). (A3)
Inside the periodic stratified medium, at any given frequency ω, the time-harmonic field ΨT(z) can be represented as a superposition
ΨT(z)=Ψk1(z)+Ψk2(z)+Ψk3(z)+Ψk4(z). 0<z<D (A4)
of four Bloch eigenmodes, each of which satisfies the following relation:
Ψk(z+L)=cikLΨk(z). (A5)
Real k correspond to propagating (traveling) Bloch modes, while complex k correspond to evanescent modes. Depending on the frequency ω, the full set of four Bloch eigenmodes in Eq. (A4) may include only propagating modes, only evanescent modes, or both. In any event, the respective set [k1,k2,k3,k4] of four Bloch wave numbers satisfies the relation:
[k1,k2,k3,k4]=[k*1,k*2,k*3,k*4]. (A6)′
Taking into account Eq. (A6), one can distinguish the following three possibilities.
(i) All four Bloch modes in Eq. (A4) are propagating
k1=k*1, k2=k*2, k3=k*3, k4=k*4. (A7)
(ii) All four Bloch modes in Eq. (A4) are evanescent. This is the case when the frequency w falls into a photonic band gap. According to Eq. (A6), one can assume that in this case
k1=k*2≠k*1, k3=k*4≠k*3. (A8)
(iii) Two of the Bloch modes in Eq. (A4) are propagating modes, while the other two are evanescent. According to Eq. (A6), one can assume that in this case
k1=k*1, k2=k*2, k3=k*4≠k*3. (A9)
In all cases, propagating modes with u>0 and evanescent modes with k″0 are referred to as forward waves. The propagating modes with u<0 and evanescent modes with k″<0 are referred to as backward waves.
In reciprocal (nonmagnetic) periodic structures, the Bloch dispersion relation ω(k) is always symmetric with respect to the points k=0 and k=π/L of the Brillouin zone
ω(k0+k)=ω(k0−k). where k0=0. π/L. (A10)
In periodic structures composed of nonbirefringent layers, every Bloch wave is doubly degenerate with respect to polarization. A typical k−ω diagram for such a case is shown in
The speed of a traveling wave in a periodic medium is determined by the group velocity u=∂ω/∂k. Normally, every spectral branch ω(k) develops stationary points (1), where the group velocity of the corresponding propagating mode vanishes. Usually, such points are located at the center and at the boundary of the Brillouin zone
ks=k0=0, π/L. (A11)
This is always the case in periodic layered structures composed of nonbirefringent layers, where all stationary points coincide with photonic band edges, as shown in
Under normal circumstances, evanescent modes decay exponentially with the distance from the periodic structure boundaries. In such cases, the evanescent contribution to ΨT can be significant only in close proximity to the surface or some other defects of the periodic structure. The situation can change dramatically in the vicinity of a stationary point (1). At first sight, stationary points (1) relate only to propagating Bloch modes. But in fact, in the vicinity of every stationary point frequency ωs, the imaginary part k″ of the Bloch wavenumber of at least one of the evanescent modes also vanishes. As a consequence, the respective evanescent mode decays very slowly, and its role may extend far beyond the photonic crystal boundary.
The final and most important remark is related to the EM eigenmodes in the vicinity of a DBE or such a RBE that is close in shape to a DBE. In either case, all four vector-columns (A1) corresponding to the four Bloch modes in Eq. (A4) are nearly parallel to each other (see the details in Refs. [22,23,33]). The latter circumstance is responsible for the giant EM field amplitude at the respective transmission resonance.
Energy Flux: Transmission and Reflection Coefficients
Let SI, SR, ST, and SP be the energy fluxes of the respective waves in
In the case of losses medium, the Poynting vector S is independent of z
S≡SI+SR=ST=SP.
In such a case t=1−r=S/SI.
In the general case of the time-harmonic EM field ΨT(z) being a superposition (A4) of several Bloch modes, propagating and/or evanescent, the contribution of each propagating mode Ψk to the total energy flux S is independent of others and can be expressed in terms of its group velocity and amplitude
Sk=Wkuk. where Wk∝<|Ψk|>2. (A13)
In the particular case of a single propagating mode, the quantity Wk in Eq. (A13) is equal to the energy density averaged over a unit cell L. By contrast, a single evanescent mode never transfers energy. Only a combination of forward and backward evanescent modes can contribute to the energy flux (see p. 327 in Ref. [22]).
Energy Density and Energy Flux at Resonance Frequency
Consider the energy flux at transmission band edge resonance formed by a pair (A15) of reciprocal Bloch waves. Assume that the amplitude of the incident wave in
ΨT(0)=Ψk(0)+Ψ−k(0)∝1.
ΨT(D)=Ψk(D)+Ψ−k(D)∝1. (A14)
According to Eq. (A13), the energy flux associated with the time-harmonic field ΨT(z) in Eq. (4) is
S=Sk+S−k≈Wku(k)+W−ku(−k)=(Wk−W−k)u(k). (A15)
where
Wk∝<|Ψk|2>. W−k∝<|Ψ−k|2>.
The conditions the finite resonance transmission is compatible with the vanishing group velocity in the vicinity of stationary point (1), where the transmission band edge resonance occurs is considered. In the vicinity of stationary point (1), the magnitude u(k) of the group velocity in Eq. (A15) is vanishingly small. The fact that the resonance energy flux remains of the order of unity implies that the amplitude of the Bloch components in Eq. (A15) increases so that
Wk−W−k∝u−1. as u→0. (A16)
In order to reconcile the boundary condition (A14) with the requirement (A16) of a finite energy flux, the following requirement on the amplitudes of the two Bloch components is imposed
Wk−W−k∝√{square root over (Wk)}∝u−1. as u→0. (A17)
The relation (A17) was derived under the assumption that the time-harmonic filed ΨT(z) inside the periodic medium is a superposition (4) of one forward and one backward propagating Bloch waves with equal and opposite wave numbers and group velocities. This is always the case for RBE related transmission resonance. However, the representation (4) is not applicable to the case of a DBE related giant resonance, because in this case the contribution of the evanescent modes is equally important [33]. Finally, in the case of SBE related transmission resonance, the relations (4) and (A17) may or may not apply, depending on whether the resonance field ΨT(z) is formed by a single pair of propagating Bloch modes, or the contribution of all four propagating modes is equally important.
Physical Characteristics of the Periodic Layered Structure Used in Numerical Simulations
The simplest periodic layered structure supporting a DBE or a SBE at normal propagation is shown in
The dielectric permittivity tensors {acute over (∈)}A in each anisotropic A layer has the form
where the parameter δ characterizes the magnitude of in-plane anisotropy, and the angle φ determines the orientation of the anisotropy axes in the x-y plane. All the A layers have the same thickness A and the same magnitude δ of inplane anisotropy. The only difference between the adjacent anisotropic layers A1 and A2 in
φ=φ1−φ2 (B3)
between the layers A1 and A2. This angle determines the symmetry of the periodic array and, eventually, the kind of k−ω diagram it can display.
In all numerical simulations related to the periodic layered structure in
∈B=16.0, ∈A=4.7797. δ=3.4572. φ=π/6. (B4)
At normal propagation, the numerical value of ∈3 in Eq. (B2) is irrelevant. The relative thickness of the A and B layers, can be different in different examples.
In all plots of the field distribution inside periodic media at 0<z<D we, in fact, plotted the following physical quantity:
∝|Ψ(z)|2>=∝{right arrow over (E)}(z)·{right arrow over (E)}*(z)+{right arrow over (H)}(z)·{right arrow over (H)}*(z) >L. (B5)
which is the squared field amplitude averaged over a local unit cell. The actual function |Ψ(z)|2, as well as the electromagnetic energy density distribution W(z), are strongly oscillating functions of the coordinate z, with the period of oscillations coinciding with the unit cell length L. Given the relation Wα|Ψ(z)|2, the quantity (B5) can also be qualitatively interpreted as the smoothed energy density distribution, with the correction coefficient of the order of unity. In all plots, the distance z, the wavenumber k, and the frequency ω are expressed in units of L, L−1, and cL−1, respectively.
In the foregoing specification, the invention has been described with reference to specific embodiments thereof. It will, however, be evident that various modifications and changes may be made thereto without departing from the broader spirit and scope of the invention. For example, the reader is to understand that the specific ordering and combination of process actions shown in the process flow diagrams described herein is merely illustrative, unless otherwise stated, and the invention can be performed using different or additional process actions, or a different combination or ordering of process actions. As another example, each feature of one embodiment can be mixed and matched with other features shown in other embodiments. Features and processes known to those of ordinary skill may similarly be incorporated as desired. Additionally and obviously, features may be added or subtracted as desired. Accordingly, the invention is not to be restricted except in light of the attached claims and their equivalents.
Figotin, Aleksandr, Vitebskiy, Ilya M.
Patent | Priority | Assignee | Title |
Patent | Priority | Assignee | Title |
5365541, | Jan 29 1992 | Northrop Grumman Systems Corporation | Mirror with photonic band structure |
5559825, | Apr 25 1995 | The United States of America as represented by the Secretary of the Army | Photonic band edge optical diode |
5689275, | May 16 1995 | Georgia Tech Research Corporation | Electromagnetic antenna and transmission line utilizing photonic bandgap material |
5751466, | Jan 11 1996 | ALABAMA, UNIVERSITY OF AT HUNTSVILLE | Photonic bandgap apparatus and method for delaying photonic signals |
5999308, | Apr 01 1998 | Massachusetts Institute of Technology | Methods and systems for introducing electromagnetic radiation into photonic crystals |
6130780, | Feb 19 1998 | Massachusetts Institute of Technology | High omnidirectional reflector |
6134043, | Aug 11 1998 | Massachusetts Institute of Technology | Composite photonic crystals |
6358854, | Apr 21 1999 | National Technology & Engineering Solutions of Sandia, LLC | Method to fabricate layered material compositions |
6414780, | Dec 23 1999 | ARMY, UNITED STATES OF AMERICA, AS REPRESENTED BY THE SECRETARY OF THE | Photonic signal reflectivity and transmissivity control using a photonic band gap structure |
6701048, | May 01 2001 | Regents of the University of California, The | Unidirectional gyrotropic photonic crystal and applications for the same |
6823111, | Jul 31 2000 | Spectalis Corp | Optical waveguide filters |
6825982, | Jun 27 2002 | The United States of Americas as represented by the Administrator of the National Aeronautics and Space Administration | Strongly-refractive one-dimensional photonic crystal prisms |
7072555, | May 01 2003 | The Regents of the University of California | Systems and methods for transmitting electromagnetic energy in a photonic device |
20010019565, | |||
20020018298, | |||
20020064343, | |||
20020162988, | |||
20040008928, | |||
20040013361, | |||
20040218651, | |||
20090315634, |
Executed on | Assignor | Assignee | Conveyance | Frame | Reel | Doc |
Oct 01 2008 | FIGOTIN, ALEKSANDR | The Regents of the University of California | ASSIGNMENT OF ASSIGNORS INTEREST SEE DOCUMENT FOR DETAILS | 027211 | /0234 | |
Oct 01 2008 | VITEBSKIY, ILYA M | The Regents of the University of California | ASSIGNMENT OF ASSIGNORS INTEREST SEE DOCUMENT FOR DETAILS | 027211 | /0234 | |
Nov 10 2011 | The Regents of the University of California | (assignment on the face of the patent) | / |
Date | Maintenance Fee Events |
Aug 18 2017 | M2551: Payment of Maintenance Fee, 4th Yr, Small Entity. |
Aug 18 2021 | M2552: Payment of Maintenance Fee, 8th Yr, Small Entity. |
Date | Maintenance Schedule |
Feb 18 2017 | 4 years fee payment window open |
Aug 18 2017 | 6 months grace period start (w surcharge) |
Feb 18 2018 | patent expiry (for year 4) |
Feb 18 2020 | 2 years to revive unintentionally abandoned end. (for year 4) |
Feb 18 2021 | 8 years fee payment window open |
Aug 18 2021 | 6 months grace period start (w surcharge) |
Feb 18 2022 | patent expiry (for year 8) |
Feb 18 2024 | 2 years to revive unintentionally abandoned end. (for year 8) |
Feb 18 2025 | 12 years fee payment window open |
Aug 18 2025 | 6 months grace period start (w surcharge) |
Feb 18 2026 | patent expiry (for year 12) |
Feb 18 2028 | 2 years to revive unintentionally abandoned end. (for year 12) |